 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2014/prelims/mathematics-i/7/a/i/solution

G

S

ϕ : G \times S \to S

satisfying

ϕ (e, s) = s, ϕ (g h, s) = ϕ (g, ϕ (h, s))

for all

s \in S

and

g, h \in G

g \cdot s = ϕ (g, s)

, these are

e \cdot s = s

and

(g h) \cdot s = g \cdot (h \cdot s)

. No commutativity is assumed, so the order of composition matters.

 New to topics? Read the docs here!